Subgroups — C₆₇×#11107 #30690358

Order 17152.

910 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#107×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#417×C₆₇	C₂	1	2	yes	maximal
3	C₂×#408×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#343×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#341×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
9	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
11	C₂×#107×C₆₇	—	2	2	no	—
12	C₂×#107×C₆₇	—	2	4	no	—
13	#417×C₆₇	—	2	4	no	—
14	#417×C₆₇	—	2	4	no	—
15	C₂×#103×C₆₇	—	2	2	no	—
16	C₂×#102×C₆₇	—	2	2	no	—
17	#408×C₆₇	—	2	2	no	—
18	#408×C₆₇	C₄	1	4	yes	complemented
19	C₂×C₂²⋊C₈×C₆₇	—	2	2	no	—
20	#343×C₆₇	—	2	4	no	—
21	#341×C₆₇	—	2	2	no	—
22	C₂×#41×C₆₇	—	2	1	no	—
23	C₂×#41×C₆₇	—	2	1	no	—
24	C₂×#41×C₆₇	—	2	2	no	—
25	C₂×#41×C₆₇	—	2	2	no	—
26	C₂×#41×C₆₇	—	2	2	no	—
27	C₂×#41×C₆₇	—	2	2	no	—
28	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
29	C₂×#41×C₆₇	Q₈	1	2	yes	—
30	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
31	C₂²×D₄×C₆₇	—	2	1	no	—
32	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₆₇	—	2	1	no	—
34	C₂³×C₄×C₆₇	—	2	2	no	—
35	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₆₇	—	2	1	no	—
37	C₂×M₁₆×C₆₇	—	2	2	no	—
38	C₂×M₁₆×C₆₇	—	2	2	no	—
39	C₂×M₁₆×C₆₇	—	2	2	no	—
40	C₂×M₁₆×C₆₇	—	2	2	no	—
41	C₂×M₁₆×C₆₇	—	2	2	no	—
42	C₂×M₁₆×C₆₇	—	2	2	no	—
43	C₂×M₁₆×C₆₇	—	2	2	no	—
44	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
45	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
46	C₂²×C₈×C₆₇	—	2	2	no	—
47	C₂²×C₈×C₆₇	—	2	2	no	—
48	C₂²×C₈×C₆₇	—	2	1	no	—
49	C₂²×C₈×C₆₇	—	2	2	no	—
50	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₆₇	D₄	1	2	yes	—
52	C₂²×C₈×C₆₇	D₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	—
59	#107×C₆₇	—	2	8	no	—
60	#103×C₆₇	—	2	8	no	—
61	#102×C₆₇	—	2	8	no	—
62	C₂²⋊C₈×C₆₇	—	2	8	no	—
63	C₂⁴×C₆₇	—	2	1	no	—
64	C₂⁴×C₆₇	#41	1	1	yes	characteristic
65	#41×C₆₇	—	2	4	no	—
66	#41×C₆₇	—	2	4	no	—
67	#41×C₆₇	—	2	4	no	—
68	#41×C₆₇	—	2	4	no	—
69	#41×C₆₇	—	4	2	no	—
70	#41×C₆₇	—	4	2	no	—
71	#41×C₆₇	—	4	2	no	—
72	#41×C₆₇	—	4	2	no	—
73	C₂×Q₈×C₆₇	—	2	1	no	—
74	C₂×Q₈×C₆₇	—	2	2	no	—
75	C₂×Q₈×C₆₇	—	2	2	no	—
76	C₂×Q₈×C₆₇	C₄⋊C₄	1	2	yes	—
77	C₂×D₄×C₆₇	—	2	2	no	—
78	C₂×D₄×C₆₇	—	2	2	no	—
79	C₂×D₄×C₆₇	—	2	2	no	—
80	C₂×D₄×C₆₇	—	2	1	no	—
81	C₂×D₄×C₆₇	—	2	2	no	—
82	C₂×D₄×C₆₇	—	4	2	no	—
83	C₂×D₄×C₆₇	—	4	2	no	—
84	C₂×D₄×C₆₇	C₄⋊C₄	1	2	yes	—
85	C₂²×C₄×C₆₇	—	2	1	no	—
86	C₂²×C₄×C₆₇	—	2	2	no	—
87	C₂²×C₄×C₆₇	—	2	2	no	—
88	C₂²×C₄×C₆₇	—	2	4	no	—
89	C₂²×C₄×C₆₇	—	4	1	no	—
90	C₂²×C₄×C₆₇	—	4	1	no	—
91	C₂²×C₄×C₆₇	—	4	1	no	—
92	C₂²×C₄×C₆₇	—	4	1	no	—
93	C₂²×C₄×C₆₇	—	4	2	no	—
94	C₂²×C₄×C₆₇	—	4	2	no	—
95	C₂²×C₄×C₆₇	—	8	4	no	—
96	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	#41	1	2	yes	—
104	C₂²×C₄×C₆₇	#41	1	2	yes	—
105	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
106	M₁₆×C₆₇	—	2	4	no	—
107	M₁₆×C₆₇	—	2	4	no	—
108	M₁₆×C₆₇	—	2	4	no	—
109	M₁₆×C₆₇	—	4	2	no	—
110	M₁₆×C₆₇	—	4	2	no	—
111	M₁₆×C₆₇	—	4	2	no	—
112	C₂×C₈×C₆₇	—	2	4	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₂×C₈×C₆₇	—	2	4	no	—
115	C₂×C₈×C₆₇	—	2	2	no	—
116	C₂×C₈×C₆₇	—	2	2	no	—
117	C₂×C₈×C₆₇	—	2	4	no	—
118	C₂×C₈×C₆₇	—	4	2	no	—
119	C₂×C₈×C₆₇	—	4	2	no	—
120	C₂×C₈×C₆₇	—	4	1	no	—
121	C₂×C₈×C₆₇	—	4	1	no	—
122	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
123	C₄⋊C₄×C₆₇	—	2	4	no	—
124	C₄⋊C₄×C₆₇	—	2	4	no	—
125	C₄⋊C₄×C₆₇	—	4	2	no	—
126	C₄⋊C₄×C₆₇	—	4	2	no	—
127	C₂³×C₆₇	—	2	2	no	—
128	C₂³×C₆₇	—	2	2	no	—
129	C₂³×C₆₇	—	2	2	no	—
130	C₂³×C₆₇	—	4	1	no	—
131	C₂³×C₆₇	—	4	1	no	—
132	C₂³×C₆₇	—	4	2	no	—
133	C₂³×C₆₇	—	4	2	no	—
134	C₂³×C₆₇	#117	1	1	yes	characteristic
135	C₂³×C₆₇	#121	1	2	yes	—
136	C₂³×C₆₇	#121	1	1	yes	characteristic
137	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
138	C₂³×C₆₇	#126	1	2	yes	—
139	Q₈×C₆₇	—	2	4	no	—
140	Q₈×C₆₇	—	4	2	no	—
141	D₄×C₆₇	—	2	4	no	—
142	D₄×C₆₇	—	4	4	no	—
143	D₄×C₆₇	—	4	4	no	—
144	D₄×C₆₇	—	4	2	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	4	1	no	—
147	C₂×C₄×C₆₇	—	4	1	no	—
148	C₂×C₄×C₆₇	—	4	2	no	—
149	C₂×C₄×C₆₇	—	4	2	no	—
150	C₂×C₄×C₆₇	—	4	2	no	—
151	C₂×C₄×C₆₇	—	4	2	no	—
152	C₂×C₄×C₆₇	—	4	4	no	—
153	C₂×C₄×C₆₇	—	8	4	no	—
154	C₂×C₄×C₆₇	—	8	4	no	—
155	C₂×C₄×C₆₇	—	8	4	no	—
156	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
159	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
160	C₂×C₄×C₆₇	#120	1	2	yes	—
161	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
162	C₂×C₄×C₆₇	#121	1	2	yes	—
163	C₂×C₄×C₆₇	#121	1	2	yes	—
164	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
165	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
166	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
167	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
168	C₂×C₄×C₆₇	#123	1	2	yes	—
169	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
170	C₂×C₄×C₆₇	#123	1	2	yes	—
171	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
172	C₂×C₄×C₆₇	#124	1	2	yes	—
173	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
174	C₈×C₆₇	—	2	4	no	—
175	C₈×C₆₇	—	4	2	no	—
176	C₈×C₆₇	—	4	4	no	—
177	C₂²×C₆₇	—	2	2	no	—
178	C₂²×C₆₇	—	2	2	no	—
179	C₂²×C₆₇	—	2	4	no	—
180	C₂²×C₆₇	—	2	2	no	—
181	C₂²×C₆₇	—	2	2	no	—
182	C₂²×C₆₇	—	4	4	no	—
183	C₂²×C₆₇	—	4	2	no	—
184	C₂²×C₆₇	—	8	2	no	—
185	C₂²×C₆₇	—	8	1	no	—
186	C₂²×C₆₇	—	8	1	no	—
187	C₂²×C₆₇	#380	1	1	yes	characteristic
188	C₂²×C₆₇	#388	1	2	yes	—
189	C₂²×C₆₇	#392	1	1	yes	characteristic
190	C₂²×C₆₇	#396	1	2	yes	—
191	C₂²×C₆₇	#397	1	1	yes	characteristic
192	C₂²×C₆₇	#472	1	2	yes	—
193	C₂²×C₆₇	#473	1	2	yes	—
194	C₄×C₆₇	—	4	2	no	—
195	C₄×C₆₇	—	8	4	no	—
196	C₄×C₆₇	#385	1	2	yes	—
197	C₄×C₆₇	#386	1	1	yes	characteristic
198	C₄×C₆₇	#390	1	1	yes	characteristic
199	C₄×C₆₇	#393	1	1	yes	characteristic
200	C₄×C₆₇	#394	1	2	yes	—
201	C₄×C₆₇	#399	1	1	yes	characteristic
202	#11107	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
203	C₂×C₆₇	—	2	2	no	—
204	C₂×C₆₇	—	2	2	no	—
205	C₂×C₆₇	—	8	2	no	—
206	C₂×C₆₇	#1439	1	1	yes	characteristic
207	C₂×C₆₇	#1816	1	2	yes	—
208	C₂×C₆₇	#2018	1	1	yes	characteristic
209	C₂×C₆₇	#2020	1	1	yes	characteristic
210	C₂×C₆₇	#2078	1	2	yes	—
211	C₂²×#107	C₂×C₆₇	1	1	yes	characteristic, complemented
212	C₂×#417	C₂×C₆₇	1	2	yes	—
213	C₂×#408	C₂×C₆₇	1	1	yes	characteristic
214	C₂×#343	C₂×C₆₇	1	2	yes	complemented
215	C₂×#341	C₂×C₆₇	1	1	yes	characteristic, complemented
216	C₆₇	#11107	1	1	yes	characteristic, direct product factor, Sylow
217	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
218	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
219	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
220	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
221	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
222	C₂×#107	—	2	2	no	—
223	C₂×#107	—	2	4	no	—
224	#417	—	2	4	no	—
225	#417	—	2	4	no	—
226	C₂×#103	—	2	2	no	—
227	C₂×#102	—	2	2	no	—
228	#408	—	2	2	no	—
229	#408	C₄×C₆₇	1	4	yes	complemented
230	C₂×C₂²⋊C₈	—	2	2	no	—
231	#343	—	2	4	no	—
232	#341	—	2	2	no	—
233	C₂×#41	—	2	1	no	—
234	C₂×#41	—	2	1	no	—
235	C₂×#41	—	2	2	no	—
236	C₂×#41	—	2	2	no	—
237	C₂×#41	—	2	2	no	—
238	C₂×#41	—	2	2	no	—
239	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
240	C₂×#41	Q₈×C₆₇	1	2	yes	—
241	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
242	C₂²×D₄	—	2	1	no	—
243	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
244	C₂³×C₄	—	2	1	no	—
245	C₂³×C₄	—	2	2	no	—
246	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
247	C₂×M₁₆	—	2	1	no	—
248	C₂×M₁₆	—	2	2	no	—
249	C₂×M₁₆	—	2	2	no	—
250	C₂×M₁₆	—	2	2	no	—
251	C₂×M₁₆	—	2	2	no	—
252	C₂×M₁₆	—	2	2	no	—
253	C₂×M₁₆	—	2	2	no	—
254	C₂×M₁₆	—	2	2	no	—
255	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
256	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
257	C₂²×C₈	—	2	2	no	—
258	C₂²×C₈	—	2	2	no	—
259	C₂²×C₈	—	2	1	no	—
260	C₂²×C₈	—	2	2	no	—
261	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
262	C₂²×C₈	D₄×C₆₇	1	2	yes	—
263	C₂²×C₈	D₄×C₆₇	1	2	yes	—
264	C₂×C₄⋊C₄	—	2	2	no	—
265	C₂×C₄⋊C₄	—	2	2	no	—
266	C₂×C₄⋊C₄	—	2	2	no	—
267	C₂×C₄⋊C₄	—	2	2	no	—
268	C₂×C₄⋊C₄	—	2	2	no	—
269	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	—
270	#107	—	2	8	no	—
271	#103	—	2	8	no	—
272	#102	—	2	8	no	—
273	C₂²⋊C₈	—	2	8	no	—
274	C₂⁴	—	2	1	no	—
275	C₂⁴	#41×C₆₇	1	1	yes	characteristic
276	#41	—	2	4	no	—
277	#41	—	2	4	no	—
278	#41	—	2	4	no	—
279	#41	—	2	4	no	—
280	#41	—	4	2	no	—
281	#41	—	4	2	no	—
282	#41	—	4	2	no	—
283	#41	—	4	2	no	—
284	C₂×Q₈	—	2	1	no	—
285	C₂×Q₈	—	2	2	no	—
286	C₂×Q₈	—	2	2	no	—
287	C₂×Q₈	C₄⋊C₄×C₆₇	1	2	yes	—
288	C₂×D₄	—	2	2	no	—
289	C₂×D₄	—	2	2	no	—
290	C₂×D₄	—	2	2	no	—
291	C₂×D₄	—	2	1	no	—
292	C₂×D₄	—	2	2	no	—
293	C₂×D₄	—	4	2	no	—
294	C₂×D₄	—	4	2	no	—
295	C₂×D₄	C₄⋊C₄×C₆₇	1	2	yes	—
296	C₂²×C₄	—	2	1	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	2	4	no	—
300	C₂²×C₄	—	4	1	no	—
301	C₂²×C₄	—	4	1	no	—
302	C₂²×C₄	—	4	1	no	—
303	C₂²×C₄	—	4	1	no	—
304	C₂²×C₄	—	4	2	no	—
305	C₂²×C₄	—	4	2	no	—
306	C₂²×C₄	—	8	4	no	—
307	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
308	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
309	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
310	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
311	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
312	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
313	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
314	C₂²×C₄	#41×C₆₇	1	2	yes	—
315	C₂²×C₄	#41×C₆₇	1	2	yes	—
316	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
317	M₁₆	—	2	4	no	—
318	M₁₆	—	2	4	no	—
319	M₁₆	—	2	4	no	—
320	M₁₆	—	4	2	no	—
321	M₁₆	—	4	2	no	—
322	M₁₆	—	4	2	no	—
323	C₂×C₈	—	2	4	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	4	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	4	no	—
329	C₂×C₈	—	4	2	no	—
330	C₂×C₈	—	4	2	no	—
331	C₂×C₈	—	4	1	no	—
332	C₂×C₈	—	4	1	no	—
333	C₂×C₈	C₂²⋊C₄×C₆₇	1	4	yes	—
334	C₄⋊C₄	—	2	4	no	—
335	C₄⋊C₄	—	2	4	no	—
336	C₄⋊C₄	—	4	2	no	—
337	C₄⋊C₄	—	4	2	no	—
338	C₂³	—	2	2	no	—
339	C₂³	—	2	2	no	—
340	C₂³	—	2	2	no	—
341	C₂³	—	4	2	no	—
342	C₂³	—	4	2	no	—
343	C₂³	—	4	1	no	—
344	C₂³	—	4	1	no	—
345	C₂³	#117×C₆₇	1	1	yes	characteristic
346	C₂³	#121×C₆₇	1	2	yes	—
347	C₂³	#121×C₆₇	1	1	yes	characteristic
348	C₂³	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
349	C₂³	#126×C₆₇	1	2	yes	—
350	Q₈	—	2	4	no	—
351	Q₈	—	4	2	no	—
352	D₄	—	2	4	no	—
353	D₄	—	4	4	no	—
354	D₄	—	4	4	no	—
355	D₄	—	4	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	4	1	no	—
358	C₂×C₄	—	4	1	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	4	4	no	—
364	C₂×C₄	—	8	4	no	—
365	C₂×C₄	—	8	4	no	—
366	C₂×C₄	—	8	4	no	—
367	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
368	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
369	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
370	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
371	C₂×C₄	#120×C₆₇	1	2	yes	—
372	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
373	C₂×C₄	#121×C₆₇	1	2	yes	—
374	C₂×C₄	#121×C₆₇	1	2	yes	—
375	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
376	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
377	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
378	C₂×C₄	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
379	C₂×C₄	#123×C₆₇	1	2	yes	—
380	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
381	C₂×C₄	#123×C₆₇	1	2	yes	—
382	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
383	C₂×C₄	#124×C₆₇	1	2	yes	—
384	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
385	C₈	—	2	4	no	—
386	C₈	—	4	2	no	—
387	C₈	—	4	4	no	—
388	C₂²	—	2	2	no	—
389	C₂²	—	2	2	no	—
390	C₂²	—	2	2	no	—
391	C₂²	—	2	2	no	—
392	C₂²	—	2	4	no	—
393	C₂²	—	4	4	no	—
394	C₂²	—	4	2	no	—
395	C₂²	—	8	1	no	—
396	C₂²	—	8	1	no	—
397	C₂²	—	8	2	no	—
398	C₂²	#380×C₆₇	1	1	yes	characteristic
399	C₂²	#388×C₆₇	1	2	yes	—
400	C₂²	#392×C₆₇	1	1	yes	characteristic
401	C₂²	#396×C₆₇	1	2	yes	—
402	C₂²	#397×C₆₇	1	1	yes	characteristic
403	C₂²	#472×C₆₇	1	2	yes	—
404	C₂²	#473×C₆₇	1	2	yes	—
405	C₄	—	4	2	no	—
406	C₄	—	8	4	no	—
407	C₄	#385×C₆₇	1	2	yes	—
408	C₄	#386×C₆₇	1	1	yes	characteristic
409	C₄	#390×C₆₇	1	1	yes	characteristic
410	C₄	#393×C₆₇	1	1	yes	characteristic
411	C₄	#394×C₆₇	1	2	yes	—
412	C₄	#399×C₆₇	1	1	yes	characteristic
413	C₂	—	2	2	no	—
414	C₂	—	2	2	no	—
415	C₂	—	8	2	no	—
416	C₂	C₆₇×#1439	1	1	yes	characteristic
417	C₂	C₆₇×#1816	1	2	yes	—
418	C₂	C₆₇×#2018	1	1	yes	characteristic
419	C₂	C₆₇×#2020	1	1	yes	characteristic
420	C₂	C₆₇×#2078	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	#121
commutator	C₂²×C₄	C₂²×C₄×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×#107×C₆₇	C₂	semidirect
4. C₂×#343×C₆₇	C₂	semidirect
5. C₂×#341×C₆₇	C₂	semidirect
18. #408×C₆₇	C₄	semidirect
202. #11107	C₆₇	direct (N × H)
211. C₂²×#107	C₂×C₆₇	semidirect
214. C₂×#343	C₂×C₆₇	semidirect
215. C₂×#341	C₂×C₆₇	semidirect
216. C₆₇	#11107	direct (N × H)
229. #408	C₄×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#107×C₆₇	2
C₂×#417×C₆₇	2
C₂×#408×C₆₇	2
C₂×#343×C₆₇	2
C₂×#341×C₆₇	2
#11107	67

Subgroups — C67×#11107 #30690358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#11107 #30690358